Chapter 11 - Rolling, Chapter 11 - Rolling, Torque, and Angular Torque, and Angular Momentum Momentum

Home , Rolling

Rolling

Angular Momentum

Conservation of Angular Momentum

“The way to catch a knuckleball is to wait until it stops rolling and then pick it up.” -Bob Uecker

David J. Starling Penn State Hazleton PHYS 211 Each velocity vector on the wheel is a sum of ~vcom and the rotation.

Chapter 11 - Rolling, Rolling Torque, and Angular Momentum

Rolling

Rolling motion is a combination of pure rotation Angular Momentum

Conservation of Angular and pure translation. Momentum Each velocity vector on the wheel is a sum of ~vcom and the rotation.

Chapter 11 - Rolling, Rolling Torque, and Angular Momentum

Rolling

Rolling motion is a combination of pure rotation Angular Momentum

Conservation of Angular and pure translation. Momentum Each velocity vector on the wheel is a sum of ~vcom and the rotation.

Chapter 11 - Rolling, Rolling Torque, and Angular Momentum

Rolling

Rolling motion is a combination of pure rotation Angular Momentum

Conservation of Angular and pure translation. Momentum Chapter 11 - Rolling, Rolling Torque, and Angular Momentum

Rolling

Rolling motion is a combination of pure rotation Angular Momentum

Conservation of Angular and pure translation. Momentum

Each velocity vector on the wheel is a sum of ~vcom and the rotation. Note: therefore, acom = Rα.

Chapter 11 - Rolling, Rolling Torque, and Angular Momentum

The relationship between the angular and linear Rolling Angular Momentum velocities is ﬁxed if the wheel does not slip. Conservation of Angular Momentum vcom = Rω Chapter 11 - Rolling, Rolling Torque, and Angular Momentum

The relationship between the angular and linear Rolling Angular Momentum velocities is ﬁxed if the wheel does not slip. Conservation of Angular Momentum vcom = Rω

Note: therefore, acom = Rα. In this case, there is a static force of friction at the pivot that does no work.

Chapter 11 - Rolling, Rolling Torque, and Angular Momentum

Rolling

If the rolling object does not slip, then the point in Angular Momentum

Conservation of Angular contact with the ﬂoor is momentarily stationary. Momentum Chapter 11 - Rolling, Rolling Torque, and Angular Momentum

Rolling

If the rolling object does not slip, then the point in Angular Momentum

Conservation of Angular contact with the ﬂoor is momentarily stationary. Momentum

In this case, there is a static force of friction at the pivot that does no work. 1 = I + MR2 ω2 2 com 1 1 = I ω2 + M(Rω)2 2 com 2 1 1 = I ω2 + Mv2 2 com 2 K = KR + KT

Chapter 11 - Rolling, Rolling Torque, and Angular Momentum

Rolling

Angular Momentum The kinetic energy of the wheel rotating about Conservation of Angular point P is Momentum

1 K = I ω2 2 p 1 1 = I ω2 + M(Rω)2 2 com 2 1 1 = I ω2 + Mv2 2 com 2 K = KR + KT

Chapter 11 - Rolling, Rolling Torque, and Angular Momentum

Rolling

Angular Momentum The kinetic energy of the wheel rotating about Conservation of Angular point P is Momentum

1 K = I ω2 2 p 1 = I + MR2 ω2 2 com 1 1 = I ω2 + Mv2 2 com 2 K = KR + KT

Chapter 11 - Rolling, Rolling Torque, and Angular Momentum

Rolling

Angular Momentum The kinetic energy of the wheel rotating about Conservation of Angular point P is Momentum

1 K = I ω2 2 p 1 = I + MR2 ω2 2 com 1 1 = I ω2 + M(Rω)2 2 com 2 K = KR + KT

Chapter 11 - Rolling, Rolling Torque, and Angular Momentum

Rolling

Angular Momentum The kinetic energy of the wheel rotating about Conservation of Angular point P is Momentum

1 K = I ω2 2 p 1 = I + MR2 ω2 2 com 1 1 = I ω2 + M(Rω)2 2 com 2 1 1 = I ω2 + Mv2 2 com 2 Chapter 11 - Rolling, Rolling Torque, and Angular Momentum

Rolling

Angular Momentum The kinetic energy of the wheel rotating about Conservation of Angular point P is Momentum

1 K = I ω2 2 p 1 = I + MR2 ω2 2 com 1 1 = I ω2 + M(Rω)2 2 com 2 1 1 = I ω2 + Mv2 2 com 2 K = KR + KT Notice how the force does not oppose the direction of motion!

Chapter 11 - Rolling, Rolling Torque, and Angular Momentum

Rolling

When a rolling object accelerates, the friction Angular Momentum

Conservation of Angular force opposes the tendency to slip. Momentum Chapter 11 - Rolling, Rolling Torque, and Angular Momentum

Rolling

When a rolling object accelerates, the friction Angular Momentum

Conservation of Angular force opposes the tendency to slip. Momentum

Notice how the force does not oppose the direction of motion! Chapter 11 - Rolling, Rolling Torque, and Angular Momentum

Rolling Lecture Question 11.1 Angular Momentum Which one of the following statements concerning a wheel Conservation of Angular Momentum undergoing rolling motion is true? (a) The angular acceleration of the wheel must be 0 m/s2. (b) The tangential velocity is the same for all points on the wheel. (c) The linear velocity for all points on the rim of the wheel is non-zero. (d) The tangential velocity is the same for all points on the rim of the wheel. (e) There is no slipping at the point where the wheel touches the surface on which it is rolling. Why is it deﬁned this way?

Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum

Rolling An object A with momentum ~p also has angular Angular Momentum Conservation of Angular momentum~l = ~r × ~p about some point O. Momentum Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum

Rolling An object A with momentum ~p also has angular Angular Momentum Conservation of Angular momentum~l = ~r × ~p about some point O. Momentum

Why is it deﬁned this way? d~r d~p = × ~p +~r × dt dt = ~v × ~p +~r × (m~a)

= ~r × ~Fnet

= ~τnet

d~l τ = net dt

Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum

From Newton’s 2nd Law, let’s take a derivative of Rolling this “momentum”: Angular Momentum Conservation of Angular Momentum

d~l d = (~r × ~p) dt dt = ~v × ~p +~r × (m~a)

= ~r × ~Fnet

= ~τnet

d~l τ = net dt

Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum

From Newton’s 2nd Law, let’s take a derivative of Rolling this “momentum”: Angular Momentum Conservation of Angular Momentum

d~l d = (~r × ~p) dt dt d~r d~p = × ~p +~r × dt dt = ~r × ~Fnet

= ~τnet

d~l τ = net dt

Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum

From Newton’s 2nd Law, let’s take a derivative of Rolling this “momentum”: Angular Momentum Conservation of Angular Momentum

d~l d = (~r × ~p) dt dt d~r d~p = × ~p +~r × dt dt = ~v × ~p +~r × (m~a) = ~τnet

d~l τ = net dt

Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum

From Newton’s 2nd Law, let’s take a derivative of Rolling this “momentum”: Angular Momentum Conservation of Angular Momentum

d~l d = (~r × ~p) dt dt d~r d~p = × ~p +~r × dt dt = ~v × ~p +~r × (m~a)

= ~r × ~Fnet d~l τ = net dt

Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum

From Newton’s 2nd Law, let’s take a derivative of Rolling this “momentum”: Angular Momentum Conservation of Angular Momentum

d~l d = (~r × ~p) dt dt d~r d~p = × ~p +~r × dt dt = ~v × ~p +~r × (m~a)

= ~r × ~Fnet

= ~τnet Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum

From Newton’s 2nd Law, let’s take a derivative of Rolling this “momentum”: Angular Momentum Conservation of Angular Momentum

d~l d = (~r × ~p) dt dt d~r d~p = × ~p +~r × dt dt = ~v × ~p +~r × (m~a)

= ~r × ~Fnet

= ~τnet

d~l τ = net dt Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum

Rolling If there is more than one particle, the total Angular Momentum angular momentum ~L is just the vector sum of the Conservation of Angular Momentum individual angular momentums~li.

N X ~L =~l1 +~l2 + ··· +~lN = ~li i=1

And the net torque on the whole system is just:

d~L ~τ = net dt Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum The angular momentum of a rigid body can be Rolling

found by adding up the angular momentum of the Angular Momentum

Conservation of Angular particles that make it up: Momentum

li = r⊥ipi = r⊥i∆mivi = r⊥i∆mi(r⊥iω) 2 = (∆mir⊥i)ω X 2 L = (∆mir⊥i)ω i ! X 2 = ∆mir⊥i ω i ~L = I~ω

Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum

Rolling

Angular Momentum

Conservation of Angular Momentum

li = r⊥i∆mivi 2 = (∆mir⊥i)ω X 2 L = (∆mir⊥i)ω i ! X 2 = ∆mir⊥i ω i ~L = I~ω

Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum

Rolling

Angular Momentum

Conservation of Angular Momentum

li = r⊥i∆mivi

= r⊥i∆mi(r⊥iω) X 2 L = (∆mir⊥i)ω i ! X 2 = ∆mir⊥i ω i ~L = I~ω

Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum

Rolling

Angular Momentum

Conservation of Angular Momentum

li = r⊥i∆mivi

= r⊥i∆mi(r⊥iω) 2 = (∆mir⊥i)ω ! X 2 = ∆mir⊥i ω i ~L = I~ω

Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum

Rolling

Angular Momentum

Conservation of Angular Momentum

li = r⊥i∆mivi

= r⊥i∆mi(r⊥iω) 2 = (∆mir⊥i)ω X 2 L = (∆mir⊥i)ω i ~L = I~ω

Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum

Rolling

Angular Momentum

Conservation of Angular Momentum

li = r⊥i∆mivi

= r⊥i∆mi(r⊥iω) 2 = (∆mir⊥i)ω X 2 L = (∆mir⊥i)ω i ! X 2 = ∆mir⊥i ω i Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum

Rolling

Angular Momentum

Conservation of Angular Momentum

li = r⊥i∆mivi

= r⊥i∆mi(r⊥iω) 2 = (∆mir⊥i)ω X 2 L = (∆mir⊥i)ω i ! X 2 = ∆mir⊥i ω i ~L = I~ω Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum

Rolling There are many similarities between linear and Angular Momentum Conservation of Angular angular momentum. Momentum Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum

Rolling

Angular Momentum

Lecture Question 11.2 Conservation of Angular Momentum What is the direction of the Earth’s angular momentum as it spins about its axis? (a) north (b) south (c) east (d) west (e) radially inward Chapter 11 - Rolling, Conservation of Angular Momentum Torque, and Angular Momentum

If the net external torque on a system is zero, the Rolling angular momentum is conserved. Angular Momentum Conservation of Angular Momentum d~L ~τ = = 0 → ~L = constant net dt Chapter 11 - Rolling, Conservation of Angular Momentum Torque, and Angular Momentum

A gyroscope is an object that spins very quickly. Rolling Such an object behaves obeys the angular Angular Momentum Conservation of Angular Newton’s Second Law. Momentum

d~L ~τ = dt Chapter 11 - Rolling, Conservation of Angular Momentum Torque, and Angular Momentum

A gyroscope is an object that spins very quickly. Rolling Such an object behaves obeys the angular Angular Momentum Conservation of Angular Newton’s Second Law. Momentum

d~L ~τ = dt Chapter 11 - Rolling, Conservation of Angular Momentum Torque, and Angular Momentum

A gyroscope is an object that spins very quickly. Rolling Such an object behaves obeys the angular Angular Momentum Conservation of Angular Newton’s Second Law. Momentum

d~L ~τ = dt d~L ~τ = dt dL = τ dt = Mgr dt dL Mgr dt dφ = = L Iω dφ Mgr = (rate of precession) dt Iω

Chapter 11 - Rolling, Conservation of Angular Momentum Torque, and Angular Momentum When a gyroscope is subjected to a gravitational Rolling

force it precesses. Angular Momentum

Conservation of Angular Momentum dL = τ dt = Mgr dt dL Mgr dt dφ = = L Iω dφ Mgr = (rate of precession) dt Iω

Chapter 11 - Rolling, Conservation of Angular Momentum Torque, and Angular Momentum When a gyroscope is subjected to a gravitational Rolling

force it precesses. Angular Momentum

Conservation of Angular Momentum

d~L ~τ = dt dL Mgr dt dφ = = L Iω dφ Mgr = (rate of precession) dt Iω

Chapter 11 - Rolling, Conservation of Angular Momentum Torque, and Angular Momentum When a gyroscope is subjected to a gravitational Rolling

force it precesses. Angular Momentum

Conservation of Angular Momentum

d~L ~τ = dt dL = τ dt = Mgr dt dφ Mgr = (rate of precession) dt Iω

Chapter 11 - Rolling, Conservation of Angular Momentum Torque, and Angular Momentum When a gyroscope is subjected to a gravitational Rolling

force it precesses. Angular Momentum

Conservation of Angular Momentum

d~L ~τ = dt dL = τ dt = Mgr dt dL Mgr dt dφ = = L Iω Chapter 11 - Rolling, Conservation of Angular Momentum Torque, and Angular Momentum When a gyroscope is subjected to a gravitational Rolling

force it precesses. Angular Momentum

Conservation of Angular Momentum

d~L ~τ = dt dL = τ dt = Mgr dt dL Mgr dt dφ = = L Iω dφ Mgr = (rate of precession) dt Iω Chapter 11 - Rolling, Conservation of Angular Momentum Torque, and Angular Momentum

Rolling

Angular Momentum Lecture Question 11.3 Conservation of Angular A solid sphere of radius R rotates about an axis that is Momentum tangent to the sphere with an angular speed ω. Under the action of internal forces, the radius of the sphere increases to 2R. What is the ﬁnal angular speed of the sphere? (a) ω/4 (b) ω/2 (c) ω (d) 2ω (e) 4ω